An own goal from Chris Wood and strikes from Sadio Mane and Roberto Firmino — his 50th for the Reds — delivered a 13th successive league win, a new club all-time high.
Jurgen Klopp's side has been victorious in all four of its opening league fixtures and is two points clear of defending champion Manchester City at the top of the table.
The UEFA Champions League holder took the lead when Trent Alexander-Arnold’s cross deflected in off Wood, and Mane added a second before the break with a clinical finish.
Firmino rounded off the scoring in the 80th minute.
Liverpool’s early play brimmed with energy as Firmino, Mohamed Salah and Mane combined with menace.
Salah twice threatened the Burnley goal. The Egypt international struck the right post with a curling effort and then brought goalkeeper Nick Pope off his line to narrow the angle.
Burnley’s defence initially stood firm as it matched the visitors’ intensity, only for a curious goal to give Liverpool the lead.
On a familiar foray forward, right-back Alexander-Arnold curled a high ball into the box that was intended for the run of Salah at the back post. However, a slight deflection off Wood’s back deceived Pope and the cross sailed over his head into the left corner of the net.
The Liverpool defender’s sheepish smile as he celebrated emphasised the fortuitous nature of the opener, ultimately deemed a Wood own goal.
Liverpool’s second shortly afterwards owed more to its ruthlessness after Ben Mee misplaced a pass which was seized upon by the press of Firmino.
The Brazil international burst forward and slid the ball into the path of Mane who calmly slipped it past Pope into the right corner of the net.
The impressive Firmino notched Liverpool's third, to rubberstamp victory, when he started and finished off a counter-attack with 10 minutes left.
Burnley has a Premier League away day when they head to Brighton on 15 September (AEST).
Liverpool is in league action on the same day when it welcomes Newcastle to Anfield. Three days later, Klopp's side begins its defence of the UEFA Champions League with a group match away to Italian outfit Napoli.